Locally Isotropic Pseudo-Riemannian Manifolds

Abstract

Locally isotropic pseudo-Riemannian manifolds are known to be locally symmetric; this result is due to Wolf. In the Riemannian setting one proof, due to Szabó, uses spectral properties of the so-called Szabó operator. In this paper we extend Szabó's method to the pseudo-Riemannian setting, obtaining results comparable to those of Wolf. Most results concerning the Szabó operator in the pseudo-Riemannian setting are presented using methods of algebraic topology. This paper exploits the polynomial nature of the Szabó operator along with its behavior over the nullcone, both of which have received little attention this far.